Currently the number of web apps (by which I mean something with a html/css/js interface that runs in a web browser) that I can run locally with my own data that I store locally (or not, my choice) is tiny. The number of native apps that give me full control of my data is huge, because native platforms make that easy to do. Local file access APIs are relatively new in the browser.
You are correct that there are plenty of tasks that are inherently about communicating with another computer, but that is not what we are talking about here. Writing a document needn't involve communicating with another computer. Editing an image needn't involve communicating with another computer. Maintaining my todo list, calculating how much time to bill a client; these needn't involve communication with another computer that I don't control.